Antisymmetric tensor

In mathematics and theoretical physics, a tensor is antisymmetric on two indices i and j if it flips sign when the two indices are interchanged:

$T_{ijk\dots} = -T_{jik\dots}$

An antisymmetric tensor is a tensor for which there are two indices on which it is antisymmetric. If a tensor changes sign under the exchange of any pair of indices, then the tensor is completely antisymmetric and it is also referred to as a differential form.

A tensor A which is antisymmetric on indices i and j has the property that the contraction with a tensor B, which is symmetric on indices i and j, is identically 0.

For a general tensor U with components $U_{ijk\dots}$ and a pair of indices i and j, U has symmetric and antisymmetric parts defined as:

$U_{(ij)k\dots}=\frac{1}{2}(U_{ijk\dots}+U_{jik\dots})$ (symmetric part)

$U_{[ij]k\dots}=\frac{1}{2}(U_{ijk\dots}-U_{jik\dots})$ (antisymmetric part)

Similar definitions can be given for other pairs of indices. As the term "part" suggests, a tensor is the sum of its symmetric part and antisymmetric part for a given pair of indices, as in $U_{ijk\dots}=U_{(ij)k\dots}+U_{[ij]k\dots}$

An important antisymmetric tensor in physics is the electromagnetic tensor F in electromagnetism.

See also

• antisymmetric matrix
• Exterior algebra
• symmetric tensor
• Levi-Civita symbol

This geometry-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Tensors Geometry stubs Wikimedia Foundation. 2010. Игры ⚽ Поможем решить контрольную работу Edward MacDowell High North Alliance Look at other dictionaries: Antisymmetric — In set theory, the adjective antisymmetric usually refers to an antisymmetric relation.The term antisymmetric function is sometimes used for odd function, although some meanings of antisymmetric are essentiality f ( y , x ) = − f ( x , y ). In… …   Wikipedia antisymmetric — I. | ̷ ̷(ˌ) ̷ ̷ ̷ ̷| adjective Etymology: anti (I) + symmetric of a square matrix or a tensor : having the sign of every element or component changed by the interchange of any two indices of that element or component II. adjective 1. : relating… …   Useful english dictionary Symmetric tensor — In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments. Symmetric tensors of rank two are sometimes called quadratic forms. In more abstract terms, symmetric tensors of general rank are… …   Wikipedia Glossary of tensor theory — This is a glossary of tensor theory. For expositions of tensor theory from different points of view, see:* Tensor * Classical treatment of tensors * Tensor (intrinsic definition) * Intermediate treatment of tensors * Application of tensor theory… …   Wikipedia Contorsion tensor — The contorsion tensor in differential geometry expresses the difference between a metric compatible affine connection with Christoffel symbol Γijk and the unique torsion free Levi Civita connection for the same metric. The contortion tensor is… …   Wikipedia Metric tensor — In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold (such as a surface in space) which takes as input a pair of tangent vectors v and w and produces a real number (scalar) g(v,w) in a… …   Wikipedia Weyl tensor — In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal… …   Wikipedia Electromagnetic tensor — Electromagnetism Electricity · …   Wikipedia Torsion tensor — In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet Serret formulas, for instance, quantifies the twist of a curve… …   Wikipedia Mathematics of general relativity — For a generally accessible and less technical introduction to the topic, see Introduction to mathematics of general relativity. General relativity Introduction Mathematical formulation Resources …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”